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amicable numbers

A pair of numbers, also known as friendly numbers, each of whose aliquot parts add to give the other number. (An aliquot part is any divisor that doesn't include the number itself).

The smallest amicable numbers are 220 (aliquot parts 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110, with a sum of 284) and 284 (aliquot parts 1, 2, 4, 71, and 142, with a sum of 220). This pair was known to the ancient Greeks, and the Arabs found several more. In 1636 Pierre de Fermat rediscovered the amicable pair 17296 and 18416; two years later René Descartes rediscovered a third pair, 9363584 and 9437056. In the 18th century Leonhard Euler drew up a list of more than 60. Then, in 1866, B. Nicoló Paganini (not the violinist!), a 16-year-old Italian, startled the mathematical world by announcing that the numbers 1184 and 1210 were friendly. This second lowest pair of all had been completely overlooked. Today, the tally of known amicable numbers has grown to about two and half million. No amicable pair is known in which one of the two numbers is a square. An unusually high proportion of the numbers in amicable pairs ends in either 0 or 5.

A happy amicable pair is an amicable pair in which both numbers are happy numbers; an example is 10572550 and 10854650.

Related entry

   • Harshad number

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